Imbibition in a model open fracture. Capillary rise, kinetic roughening, and intermittent avalanche dynamics

Abstract

The dynamics of fractured media flows is relevant in many processes of interest that range from the micrometre to the kilometre lengthscales. Flow in disordered media has been proven to be an interesting system to study fundamental physics problems also. The goal of this Thesis is to study the spatio-temporal dynamics of the oil-air interface between displaced air and invading oil, in imbibition displacements through a model open fracture. The research combines exhaustive experimental work with accurate data analysis based on methods of nonlinear statistical physics. Imbibition is a process of fluid transport in a medium in which an invading fluid that preferentially wets the medium displaces the previously resident, immiscible fluid. According to the driving protocols we define spontaneous imbibition, in which the interface is driven at constant pressure difference between the inlet and the outlet of the medium and the flow rate is free to change in time, and forced-flow imbibition in which a constant flow rate of fluid is imposed at the inlet and the pressure difference may change in time. Our model open fracture consists of a Hele-Shaw (HS) cell, i.e. two parallel plates separated by a narrow gap spacing. Two configurations that mimic an open fracture have been explored: a flat HS cell, with constant aperture, and a HS cell with a dichotomic gap spacing randomly distributed in space (disordered cell). Silicone oils of different viscosities have been used as invading fluid. The advancement of the fluid front is recorded by using either CCD or CMOS cameras. An edge-tracking algorithm is applied to the binarized images to obtain front positions. We first study the evolution in time of the mean position of the interface in spontaneous imbibition experiments (capillary rise) through our two model open fractures. Experiments are performed with and without the presence of an effective gravity, achieved systematically tilting the cell against the advancement of the front or keeping it horizontal. Different pressure differences between the inlet and the outlet are systematically explored as well. We propose a new analytical solution for the spatially-averaged position of the imbibing front, based on a pressure balance equation, that reproduces experimental results at all times. In invasion of the disordered cell, capillary pressure and permeability variations distort imbibition fronts due to medium heterogeneities, while viscous pressure and surface tension tend to restore their flatness. As a result, the oil-air interface develops long-range correlations, with a lateral correlation length that depends on the capillary number Ca, tuned experimentally. Consequently, fronts advancing through the disordered cell are not flat during the whole experiment but get rough as the fluid penetrates the medium from an initially-flat interface to a final, statistically-invariant rough front. The kinetic roughening process, that occurs as a consequence of the competition of forces acting on the interface at different lengthscales, has been characterized in low-viscosity, forced-flow imbibition displacements obtaining a super-rough scaling scenario. The complex spatio-temporal dynamics of the front is studied at the statistically-stationary state of saturated front roughness in forced-flow experiments. We have analysed the spatial and temporal correlations of velocities of the front from the local scale, much smaller than the lateral correlation length and the characteristic length of the disorder, to the system size. Imbibition fronts exhibit burst-like dynamics, advancing by spatially-localized avalanches. These avalanches are power-law distributed in sizes and durations with exponential cutoffs. Power-law exponents are independent of the experimental conditions while the cutoffs diverge as Ca is reduced. We study also the intermittent character of these displacements by analysing different moments of the statistical distributions of velocity increments as a function of the time lag. We show that intermittency is controlled by two parameters only. The ensemble of results presented in this Thesis supports a very general picture of the nonequilibrium dynamics of slowly-driven fronts in open fractures. The lateral propagation of interfacial fluctuations is controlled by local mass conservation, through the lateral correlation length. The advancement of the interface in the direction of propagation is controlled by the characteristic extent of the disorder and by the mean front velocity.

L'objectiu de la tesi és l'estudi de la dinàmica espacio-temporal de la interfície entre aire desplaçat i oli invasor, en desplaçaments d'imbibició a través d'un model de fractura oberta. La recerca presentada combina un extens i exhaustiu treball experimental amb una anàlisi de dades acurada, basada en mètodes utilitzats en física estadística de no-equilibri.

El procés d'imbibició, en que el fluid invasor mulla preferentment el medi envaït, és rellevant en diverses situacions d'interès, des de fluxos fisiològics a la irrigació del sòl i l'extracció de petroli. També és un sistema model interessant per a l'estudi de problemes de física fonamental degut a les correlacions de llarg abast que es desenvolupen al front, que indueixen una dinàmica complexa.

Primer s'estudia l'avançament de la posició mitjana del front de fluid en condicions d'imbibició espontània (ascens capil•lar). Hem proposat una nova solució analítica que reprodueix els resultats experimentals tant amb presència de gravetat efectiva oposant-se a l'avançament del fluid com sense.

En experiments d'imbibició forçada s'ha caracteritzat el procés d'arrugament dinàmic (kineticroughening) del front oli-aire a baixa viscositat. L'escenari d'escalament observat és super-rugós.

Finalment s'ha estudiat la dinàmica del sistema en el règim estadísticament estacionari. S'han analitzat les correlacions temporals i espacials de les velocitats des de l'escala local, per sota la mida de les heterogeneïtats del desordre, fins a la mida del sistema. El front mostra una dinàmica a batzegades caracteritzada en termes d'allaus. Les mides i durades d'aquestes allaus estan distribuïdes estadísticament en llei de potències, amb exponents independents de les condicions experimentals, amb un truncament exponencial, que divergeix en reduir el nombre de capil•laritat.

La intermitència del senyal s'ha quantificat i se n'ha extret els dos paràmetres que la controlen.

El conjunt de resultats presentats en aquesta tesi dóna suport a una descripció molt general de la dinàmica de propagació lenta de fronts d'imbibició fora de l'equilibri en fractura oberta. La conservació local de massa controla la correlació lateral de les fluctuacions de la interfície. La longitud característica de les illes de desordre i la velocitat mitjana del front, per la seva banda, controlen l'avançament del front en la direcció de propagació.